Units, Physical Quantities, and Vectors

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PowerPoint®_{Lectures for}

University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson

Chapter 1

Units, Physical

Quantities, and Vectors

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Goals for Chapter 1

• To learn three fundamental quantities of physics and the

units to measure them

• To keep track of significant figures in calculations

• To understand vectors and scalars and how to add vectors

graphically

• To determine vector components and how to use them in

calculations

• To understand unit vectors and how to use them with

components to describe vectors

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The nature of physics

• Physics is an experimental science in which

physicists seek patterns that relate the phenomena of

nature.

• The patterns are called physical theories.

• A very well established or widely used theory is

called a physical law or principle.

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Solving problems in physics

• A problem-solving strategy offers techniques for setting up

and solving problems efficiently and accurately.

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Standards and units

• Length, time, and mass are three fundamental

quantities of physics.

• The International System (SI for Système

International) is the most widely used system of

units.

• In SI units, length is measured in meters, time in

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Unit prefixes

• Table 1.1 shows some larger and smaller units for the

fundamental quantities.

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Unit consistency and conversions

• An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re

adding “apples to apples.”)

• Always carry units through calculations.

• Convert to standard units as necessary. (Follow Problem-Solving Strategy 1.2)

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Uncertainty and significant figures—Figure 1.7

• The uncertainty of a measured quantity is indicated by its number of

significant figures.

• For multiplication and division, the answer can have no more significant figures than the smallest number of significant figures in the factors.

• For addition and subtraction, the number of significant figures is determined by the term having the

fewest digits to the right of the decimal point.

• Refer to Table 1.2, Figure 1.8, and Example 1.3.

• As this train mishap illustrates, even a small percent error can have

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Estimates and orders of magnitude

• An order-of-magnitude estimate of a quantity

gives a rough idea of its magnitude.

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Vectors and scalars

• A scalar quantity can be described by a single

number.

• A vector quantity has both a magnitude and a

direction in space.

• In this book, a vector quantity is represented in

boldface italic type with an arrow over it: A.

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Drawing vectors—Figure 1.10

• Draw a vector as a line with an arrowhead at its tip.

• The length of the line shows the vector’s magnitude.

• The direction of the line shows the vector’s direction.

• Figure 1.10 shows equal-magnitude vectors having the same

direction and opposite directions.

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Adding two vectors graphically—Figures 1.11–1.12

• Two vectors may be added graphically using either the parallelogram method or the head-to-tail method.

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Adding more than two vectors graphically—Figure 1.13

• To add several vectors, use the head-to-tail method.

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Subtracting vectors

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Multiplying a vector by a scalar

• If c is a scalar, the

product cA has

magnitude |c|A.

• Figure 1.15 illustrates

multiplication of a vector

by a positive scalar and a

negative scalar.

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Addition of two vectors at right angles

• First add the vectors graphically.

• Then use trigonometry to find the magnitude and direction of the sum.

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Components of a vector—Figure 1.17

• Adding vectors graphically provides limited accuracy. Vector components provide a general method for adding vectors.

• Any vector can be represented by an x-component A_x and a y-component A_y.

• Use trigonometry to find the components of a vector: A_x = Acos θ and A_y = Asin θ, where θ is measured from the +x-axis toward the +y-axis.

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Positive and negative components—Figure 1.18

• The components of a vector can be positive or negative numbers, as shown in the figure.

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Finding components—Figure 1.19

• We can calculate the components of a vector from its magnitude and direction.

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Calculations using components

• We can use the components of a vector to find its magnitude

and direction:

• We can use the components of a

set of vectors to find the components

of their sum:

• Refer to Problem-Solving

Strategy 1.3.

2 2_{and tan} y x y x A A A A A , x x x x y y y y R A B C R A B C

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Adding vectors using their components—Figure 1.22

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Unit vectors—Figures 1.23–1.24

• A unit vector has a magnitude

of 1 with no units.

• The unit vector î points in the

+x-direction, points in the

+y-direction, and points in the

+z-direction.

• Any vector can be expressed

in terms of its components as

A =A

_x

î+ A

_y

+ A

_z

.

• Follow Example 1.9.

jj kk jj k k

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The scalar product—Figures 1.25–1.26

• The scalar product (also called the “dot product”) of two vectors is • Figures 1.25 and 1.26 illustrate the scalar product.

cos . AB A B

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Calculating a scalar product

[Insert figure 1.27 here]

• In terms of components,

• Example 1.10 shows how to calculate a scalar product in two ways. . z z x x y y A B A B A B A B

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Finding an angle using the scalar product

• Example 1.11 shows how to use components to find the angle between two vectors.

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The vector product—Figures 1.29–1.30

• The vector

product (“cross product”) of two vectors has magnitude

and the

right-hand rule gives

its direction. See Figures 1.29 and 1.30.

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Calculating the vector product—Figure 1.32

• Use ABsin to find the

magnitude and the right-hand rule to find the direction.